Warning: Undefined array key "HTTP_HOST" in /Users/a1184615/Documents/Github.io/ProfAJRoberts/Sites/header.PHP on line 3 Construct invariant manifolds such as centre manifolds, spectral submanifolds, nonlinear normal modes, of ordinary, partial, or delay differential equations (autonomous or periodically forced)

Overview

I provide a procedure to construct a specified invariant manifold for a specified system of ordinary differential equations or delay differential equations or partial differential equations. The invariant manifold may be any of a centre manifold, a slow manifold, an un/stable manifold, a sub-centre manifold, a nonlinear normal form, or any spectral submanifold. Thus the procedure may be used to analyse pitchfork bifurcations, or oscillatory Hopf bifurcations, or any more complicated superposition, or multiscale patterns in PDEs. In the cases when the neglected spectral modes all decay, the constructed invariant manifold supplies a faithful, large time, emergent, model of the dynamics of the differential equations. Further, in the case of a slow manifold, this procedure now derives vectors defining the projection onto the invariant manifold along the isochrons: this projection is needed for initial conditions, forcing, system modifications, and uncertainty quantification.

The procedure now also empowers one to account for sinusoidal time dependence in the ODEs or PDEs, such as to derive spectral sub-manifold models of forced nonlinear normal modes.

In the case of PDEs

The analysis assumes that the 'spatial' gradients are small, that is, the spatial variations are gradual in space. The manifold is then multiscale. Currently caters for up to three `spatial' variables.

Execute on your computer?

The procedure uses computer algebra, the package Reduce, to construct approximations to the invariant manifolds.
  1. So download and install Reduce, and then download InvariantManifold.zip.
  2. Startup Reduce in the unzipped  InvariantManifold  folder.
  3. Execute  in_tex "invariantManifold.tex"$  to load the procedure.
  4. Test by executing  exampleslowman();  and confirm the output is as in  invariantManifold.pdf 
  5. See examples in  diverseExamples.pdf  and then try for systems of your interest.

Maybe use the web service  Alternatively, click this link to expand this page. Then via the web form below you may obtain an invariant manifold of your specified system of ordinary differential equations (ODE) or delay differential equations (DDE), when the ODE/DDE has fast and centre modes. The modes may be slow, as in a pitchfork bifurcation, or oscillatory, as in a Hopf bifurcation, or some more complicated superposition. In the case when the fast modes all decay, the centre manifold supplies you with a faithful large time model of the dynamics.

For example, this web page could help you analyse the long time dynamics of the system \[ \frac{d\vec u}{dt} =\left[\begin{array}{ccc} 2&1&2\\ 1&-1&1\\ -3&-1&-3 \end{array}\right] \vec u +\varepsilon \left[\begin{array}{c}u_2u_3\\ -u_1u_3\\ -u_1u_2\end{array}\right]. \] As this system is already entered for you, just enter the magic word, then click on the Submit button to see.

The bottom of the web form lists further examples.

For example, you can obtain the equivalent modulation equations corresponding to a given set of ODE/DDEs that have one or more oscillatory modes (a Hopf bifurcation for example). The analysis provides you with equations for the evolution of the complex amplitudes of the oscillators. This approach is better than averaging/homogenisation.

FYI: the latest source code is now available for use and collaborative development via the folder InvariantManifold of a Github repository

Non-autonomous ODEs?

Systems with slowly-varying time-dependence, or with sinusoidal time-dependence may be analysed here. Systems with more general time-dependence are significantly more difficult, but are analysed via the web page Normal form of stochastic or deterministic multiscale differential equations.

If you like this web page, please link to it so others can find it more easily.

Copyright Tony Roberts, School of Mathematical Sciences, University of Adelaide, South Australia. ORCID=0000-0001-8930-1552 14-October-2025 23:15